Function Norms

Definition: Function norms are norm definitions applied to function spaces. Function spaces are vector spaces whose elements are functions. The most common function spaces are $L^p$ spaces.

$$V = \{f(.) | f: [0,1] \rightarrow \mathbb{R} s.t. \int_0^1 |f(x)|^p dx < \infty \space \space \space 1 \leq p < \infty\}$$

We can define norms on $V$ as follows:

$$\|f\|_p := \left( \int_0^1 |f(x)|^p dx \right)^{1/p}$$

where $p \geq 1$.

Specific Cases

• $\|f\|_1 = \int_0^1 |f(x)| dx$ $L_1-norm$
• $\|f\|_2 = \left( \int_0^1 |f(x)|^2 dx \right)^{1/2}$ $L_2-norm$
• $\|f\|_{\infty} = \sup_{x \in [0,1]} |f(x)|$ $L_{\infty}-norm$

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#EE501 - Linear Systems Theory at METU